System Of Linear Integral Equations Research Articles

M. M. Lavrentiev-type systems and reconstructing parameters of viscoelastic media

Abstract We consider a nonlinear coefficient inverse problem of reconstructing the density and the memory matrix of a viscoelastic medium by probing the medium with a family of wave fields excited by moment tensor point sources. A spatially non-overdetermined formulation is investigated, in which the manifolds of point sources and detectors do not coincide and have a total dimension equal to three. The requirements for these manifolds are established to ensure the unique solvability of the studied inverse problem. The results are achieved by reducing the problem to a chain of connected systems of linear integral equations of the M. M. Lavrentiev type.

Spectral-Galerkin method for second kind VIEs with highly oscillatory kernels of the stationary point

In this paper, we discuss an efficient spectral approach for solving a class of second kind VIEs with highly oscillatory kernels possessing the stationary point. First, we use one variable transform to convert the highly oscillatory problem into the long-time one, and then split the long-time problem into a linear system of integral equations by using a dilation approach. Next on each interval we study the characteristic of the original solution and then propose an efficient spectral-Galerkin method for solving the linear system of integral equations. We prove that the proposed algorithm requires to solve the number O(log2⁡ω) of linear algebra equations of order n+1 and reaches the convergence order O(n−r), where r denotes the degree of the regularity of the original solution and ω denotes the wave number. Moreover, we give the approach of computing the highly oscillatory integrals produced in the spectral-Galerkin method. At last, two numerical examples are provided to verify the efficiency of our proposed method.

Characterization of the Mean First-Passage Time Function Subject to Advection in Annular-like Domains

Cell migration in a biological medium towards a blood vessel is modeled, as a random process, sucessively inside an annulus (two-dimensional domain) and an annular cylinder (three-dimensional domain). The conditional probability function u for the cell moving inside such domains (tissue) fulfills by assumption a diffusion–advection equation that is subject to a Dirichlet boundary condition on the outer boundary and a Robin boundary condition on the inner boundary. The mean first-passage time (MFPT) function determined by u estimates the average time for the travelling cell to reach various interesting targets. The MFPT function fulfills a Poisson equation inside a domain with suitable boundary conditions, which give rise to various mathematical problems. The main novelty of this study is the characterization of such an MFPT function inside an annulus and an annular cylinder, which is subject to a Robin boundary condition on the inner boundary and a Dirichlet boundary condition on the outer one, and these are integral functions whose densities are the solution of an inhomogeneous system of linear integral equations.

A Refinement Sum-Technique in an Iterative Scheme Adapted for a Linear System of Integral Equations to Approach a Fredholm Integral Equation’s Solution

A Refinement Sum-Technique in an Iterative Scheme Adapted for a Linear System of Integral Equations to Approach a Fredholm Integral Equation’s Solution

Co-inversion of a scattering cavity and its internal sources: uniqueness, decoupling and imaging

This paper concerns the simultaneous reconstruction of a sound-soft cavity and its excitation sources from the total-field data. Using the single-layer potential representations on two measurement curves, this co-inversion problem can be decoupled into two inverse problems: an inverse cavity scattering problem and an inverse source problem. This novel decoupling technique is fast and easy to implement since it is based on a linear system of integral equations. Then the uncoupled subproblems are respectively solved by the modified optimization and sampling method. We also establish the uniqueness of this co-inversion problem and analyze the stability of our method. Several numerical examples are presented to demonstrate the feasibility and effectiveness of the proposed method.

Численное решение дифференциально-алгебраических уравнений произвольного индекса с производной Римана-Лиувилля

In the article, linear systems of ordinary differential equations of fractional order α∈(0,1) are investigated. In contrast to previously known results, the authors consider the case when the matrix before the fractional differentiation operation is degenerate. Problems in such a formulation are called differential-algebraic equations of fractional order. The fundamental differences of such systems from the classical problems of fractional differentiation and integration are emphasized, namely, the systems under consideration can have an infinite number of solutions, or a solution of the original problem depends on the high fractional derivative of the right-hand side. Corresponding examples are given. The authors pass to a different, equivalent formulation of the problem, namely, they rewrite it in the form of a system of linear integral equations of the Abel type (with a weak singularity). This technique allows one to use the apparatus of regular matrix bundles to investigate the existence and uniqueness of the original problem. Using this result, the authors give sufficient conditions for the existence of a unique solution to the class of problems under consideration. Further, an algorithm for the numerical solution of such equations is proposed. The method is based on the product integration method and the quadrature formula of right rectangles. Calculations and graphs of the errors of the proposed method for various fractional differentiation exponents and various indices of the initial matrix bundles are presented.

On Solvability Conditions for a Certain Conjugation Problem

We study a certain conjugation problem for a pair of elliptic pseudo-differential equations with homogeneous symbols inside and outside of a plane sector. The solution is sought in corresponding Sobolev–Slobodetskii spaces. Using the wave factorization concept for elliptic symbols, we derive a general solution of the conjugation problem. Adding some complementary conditions, we obtain a system of linear integral equations. If the symbols are homogeneous, then we can apply the Mellin transform to such a system to reduce it to a system of linear algebraic equations with respect to unknown functions.

On the solvability of the tracking problem in the optimization of the thermal process by moving point controls

In the present article we investigate problems of tracking in the moving point control of thermal processes described by Fredholm integro-differential equations in partial derivatives with the Fredholm integral operator, in the case when the functions of point sources are nonlinear with respect to the control function. It is found that optimal controls are defined as solutions to a system of linear integral equations, and an algorithm for constructing its solution is developed. Sufficient conditions for the unique solvability of the tracking problem are found and an algorithm for constructing a complete solution to the nonlinear optimization problem was indicated.

A New Solution for The Enzymatic Glucose Fuel Cell Model with Morrison Equation via Haar Wavelet Collocation Method

Integral equations are essential tools in various areas of applied mathematics. A computational approach to solving an integral equation is important in scientific research. The Haar wavelet collocation method (HWCM) with operational matrices of integration is one famous method which has been applied to solve systems of linear integral equations. In this paper, an approximated analytical method based on the Haar wavelet collocation method is applied to the system of diffusion convection partial differential equations with initial and boundary conditions. This system determines the enzymatic glucose fuel cell with the chemical reaction rate of the Morrison equation. The enzymatic glucose fuel cell model describes the concentration of glucose and hydrogen ion that can be converted into energy. During the process, the model reduces to the linear integral equation system including computational Haar matrices. The computational Haar matrices can be computed by HWCM coding in the Maple program. Illustrated examples are provided to demonstrate the preciseness and effectiveness of the proposed method. The results are shown as numerical solutions of glucose and hydrogen ion.

Numerical Modeling of Solutions of the Applied Problems for the Poisson’s Equation

Many problems of continuum physics, applied physics, electrostatics, heat engineering, and structural mechanics are described by boundary value problems for the Poisson equation. The article is devoted to the construction of a numerical algorithm for solving plane boundary value problems for the Poisson equation. The algorithm is based on the transition to a polyharmonic equation, which is solved by the method of linear boundary elements. We consider two boundary value problems with different boundary conditions: Dirichlet boundary conditions and Neumann boundary conditions. To construct a numerical solution using the linear boundary element method, the boundary of the area is replaced by an inscribed polygon. The boundary conditions are satisfied at the middle (control) points of the elements. A polyharmonic equation is reduced to a system of linear integral equations, which can be reduced to a system of linear algebraic equations. Test numerical examples confirming the effectiveness of the proposed algorithm are given.

A graph in the form of a triangle with attached outgoing edges at each vertex

We study one incoming, two outgoing and triangle graphs for the equation of linear KdV. Using the theory of potentials, we reduce the problem to systems of linear integral equations and show that they are uniquely solvable under conditions of the uniqueness theorem.

An optimal maintenance strategy for multi-state deterioration systems based on a semi-Markov decision process coupled with simulation technique

Maintenance optimization of multi-state systems is a research topic of practical significance for various manufacturing systems. Nevertheless, design of satisfactory maintenance strategies remains a challenging problem due to the complex industrial field and large strategy space for maintenance optimization of such systems. In this study, a model of the deterioration process of multi-state systems with the multi-phase property was constructed by utilizing semi-Markov decision theory and linear systems of integral equation. An efficient maintenance strategy was developed with the purpose of maximizing the net revenue per unit period based on the simulation technique. In the constructed model, a system of linear integral equations with a recursive structure was constructed to obtain all possible transition probabilities between the states. Due to this recursive structure, conventional solutions of linear integral equations are not capable of solving this equation. For this reason, an algorithm was developed that was instrumental in resolving this issue. In the designed maintenance strategy, to avoid the potential curse of dimensionality resulting from the scales of the systems and the strategy space, a semi-Markov decision process coupled with the simulation technique was adopted in each decision period to reduce the search space by eliminating unreachable states. Finally, numerical examples of two systems with different scales based on real data of the inspection record database at Dongbei Special Steel Group Co., Ltd. are given to illustrate the effectiveness of the proposed model and the performance of the designed strategy. The numerical results demonstrate the high efficiency of the optimal strategy, especially for large-scale systems.

Dynamic model of real investment with lags

The dynamic optimal control problem of corporation’s real investment in case of a limited budget is analyzed. Initial optimization problem is formulated as a system of linear integral equations of Volterra type. The offered model takes into account the accumulated effect of real investment and previous revenue on sales. It is supposed that the previous revenue indirectly affects the current one and reflects the influence of other factors. The accumulated impact of investment is distributed over a fix time interval. The total profit maximization problem under restrictions is stated. The existence theorem for the solutions of the total profit maximization problem is proved. The model is tested with the real data. In this case the Pontryagin’s maximum principle and numerical methods of solving the integral and differential equations are used. The results of experiment are in accordance with the understanding of optimal investment strategy. Considering it the authors state that the offered model makes a solution which is possible to increase the total profit over the planning period.

Direct and inverse scattering problems for a first-order system with energy-dependent potentials

The direct and inverse scattering problems on the full line are analyzed for a first-order system of ordinary linear differential equations associated with the derivative nonlinear Schrödinger equation and related equations. The system contains a spectral parameter and two potentials, where the potentials are proportional to the spectral parameter and hence are called energy-dependent potentials. Using the two potentials as input, the direct problem is solved by determining the scattering coefficients and the bound-state information consisting of bound-state energies, their multiplicities, and the corresponding norming constants. Using two different methods, the corresponding inverse problem is solved by determining the two potentials when the scattering data set is used as input. The first method involves the transformation of the energy-dependent system into two distinct energy-independent systems. The second method involves the establishment of the so-called alternate Marchenko system of linear integral equations and the recovery of the energy-dependent potentials from the solution to the alternate Marchenko system.

An iterative method to compute minimum norm solutions of ill-posed problems in Hilbert spaces

We study an algorithm to compute minimum norm solution of ill-posed problems in Hilbert spaces and investigate its regularizing properties with discrepancy principle stopping rule. This algorithm results from straightly applying the LSQR method to the main problem before discretizing. In fact, the proposed algorithm obtains a sequence of approximate solutions of the original problem. In order to test the new algorithm, it is implemented to solve system of linear integral equations of the first kind and some examples are given. Moreover, we compare the presented algorithm with the Tikhonov regularization method to compute the least norm solution when there are more than one solution.

Pseudo-differential equations and conical potentials: 2-dimensional case

We consider two-dimensional elliptic pseudo-differential equation in a plane sector. Using a special representation for an elliptic symbol and the formula for a general solution we study the Dirichlet problem for such equation. This problem was reduced to a system of linear integral equations and then after some transformations to a system of linear algebraic equations. The unique solvability for the Dirichlet problem was proved in Sobolev-Slobodetskii spaces and a priori estimate for the solution is given.

Reliable numerical algorithm for handling fuzzy integral equations of second kind in Hilbert spaces

Integral equations under uncertainty are utilized to describe different formulations of physical phenomena in nature. This paper aims to obtain analytical and approximate solutions for a class of integral equations under uncertainty. The scheme presented here is based upon the reproducing kernel theory and the fuzzy real-valued mappings. The solution methodology transforms the linear fuzzy integral equation to crisp linear system of integral equations. Several reproducing kernel spaces are defined to investigate the approximate solutions, convergence and the error estimate in terms of uniform continuity. An iterative procedure has been given based on generating the orthonormal bases that rely on Gram-Schmidt process. Effectiveness of the proposed method is demonstrated using numerical experiments. The gained results reveal that the reproducing kernel is a systematic technique in obtaining a feasible solution for many fuzzy problems.

Thermal stresses in axisymmetric truncated conical shell with piecewise constant heat transfer coefficients

The method of solving the system of thermoelasticity equations in displacements for a thin truncated conical shell with a piecewise constant coefficients of heat transfer on the front surfaces is given. The method consists in the reduction of the system of differential equations to the system of linear integral equations of the second kind. The influence of variable coefficients of heat transfer on deflection and temperature stresses in a shell is investigated. Cite as: A. I. Chyzh, “Thermal stresses in axisymmetric truncated conical shell with piecewise constant heat transfer coefficients,” Prykl. Probl. Mekh. Mat. , Issue 16, 132–140 (2018) (in Ukrainian), http://doi.org/10.15407/apmm2018.16.132-140

A study of normality and continuity for mixed integral equations

This paper presents a numerical method for solving a mixed Fredholm–Volterra linear integral equation of the second kind in a Banach space. Under certain conditions, the existence and uniqueness of the solution are proved, using Banach’s fixed point theorem. Using Nystrom’s method, the problem is reduced to a system of linear integral equations, whose solution is then found by the resolvent method. The ideas are interesting and this area caught the attention of many researchers, having so many applications. This paper starts with a brief introduction in the subject and then proposes a new scheme which is discussed in details. The numerical examples in Sect. 6 illustrate the applicability of the theoretical results.

Sign-Definiteness of the Integral Cost on a System of Linear Integral Equations of Volterra Type

We study a question of sign-definiteness of a quadratic cost subject to the system of linear integral equations which has no Legendre term, i.e. is totally degenerate. Therefore, it cannot be studied by the clasical theory of quadratic forms. However, applying some generalizations of the known Goh transform, it is possible to reduce the given functional to a nondegenerate one, and hence, to obtain new necessary conditions of its nonnegativity. Here we reduce the original problem to a form which differs from the “classical” one only by parameter which defines endpoint value of a new control, by using Goh transformation (Goh (1966)). Thus, a new class of problems with reduced cost is investigated and corresponding necessary optimality conditions are obtained.